The Properties Of Solutions Of Complex Differential-difference Equations And The Zero Distribution Of Complex Differential-difference Polynomials

Nevanlinna theory of meromorphic function and its corresponding difference simulation theory are mainly applied to study the existence and growth of the solution of the Fermat type differential-difference equation new roman.fd New Roman.fd new roman.fdx、the property of the exponential type polynomial solution of the differential-difference equation and the zero distribution of the differential difference polynomial.The thesis is divided into five chapters.At first,the progress of solutions of complex differential equations、complex difference equations and complex differential-difference equations are introduced.In addition,the relevant notations and some basic results of Nevanlinna theory are introduced in the Chapter 1.The existence of solutions of Fermat type difference equations f(z)2+α(z)2(eP(z)2f(z+c)2=Q(z)and f(z)2+(eP(z)2f(z+c)2=eλz are studied by using Hadmard factorization theorem in the Chapter 2,in which these equations has no transcendental entire solutions with finite order.The propertie of solutions of Fermat type differential-difference equations f(z)2+(eP(z)2f(z+c)2=Q(z)eα(z)are studied in the Chapter 3.If f is a transcendental entire solutions with finite order,then the form of f is obtained.In the end,we also consider the growth of f(z)2+f(k)(z)f(z+c)-1,in which the order of growth is at least 1.The growth of solutions of non-linear differential-difference equation fn(z)+q(z)eQ(z)f(k)(z+c)=p1eα1z+p2eα2z and fn(z)+q(z)eQ(z)Δcf=pieλz+p2e-λz is invwstigated in the Chapter 4,where n≥1 and k≥1 are two integers,q(z)is a non-zero polynomial and Q(z)is a non-constant polynomial.c,λ,α1,α2,p1 and p2 be non-zero constants,α1≠α2.In particular,we show that exponential polynomial solutions satisfying certain conditions must reduce to rather specific forms,which is an improvement of previous results.The zero distribution of differential-difference polynomials is studied by using the second main theorem of Nevanlinna theory and Hadmard factorization theorem in the Chapter 5,if n≥2,then fn(z)(f’(z)+f(z+c))-b(z)has infinitely many zeros.In addition,the zero distribution of some type difference polynomials is also studied,which general results are shown.